{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Linear regression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 5.09353825e-05],\n",
       "       [ 2.12425850e-01],\n",
       "       [ 2.27277367e+01],\n",
       "       [ 1.07168045e-01],\n",
       "       [-6.53624164e+01],\n",
       "       [-5.79337497e+00]])"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "from numpy.linalg import pinv\n",
    "\n",
    "X = np.array([\n",
    "    [1, 2104, 5, 2104, 1, 45],\n",
    "    [1, 1416, 3, 2104, 2, 40],\n",
    "    [1, 1534, 3, 2104, 2, 30],\n",
    "    [1, 852, 2, 2104, 1, 36],\n",
    "])\n",
    "\n",
    "y = np.array([\n",
    "    [460],\n",
    "    [232],\n",
    "    [315],\n",
    "    [178],\n",
    "])\n",
    "\n",
    "XT = np.transpose(X)\n",
    "\n",
    "pinv(XT @ X) @ XT @ y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Noninvertibility\n",
    "\n",
    "Q: How do we calculate the inverse of $X^TX$ if it is non-invertible (singular/degenerate)?\n",
    "\n",
    "A: `pinv` will do it for you (pseudo-inverse)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0., 0.],\n",
       "       [0., 0.]])"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.zeros((2, 2))\n",
    "\n",
    "Inverse = pinv(A)\n",
    "Inverse"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Q: When would it be non-invertible?\n",
    "\n",
    "A: A few causes:\n",
    "\n",
    "1. Redundant features (linearly dependent)\n",
    "  - Example: $x_1$ = size in feet squared and $x_2$ = size in meters squared\n",
    "1. Too many features (example: $m \\leq n$)\n",
    "  - Example: $m=10$ (10 training set items) but $n=100$ (100 features)\n",
    "  - To solve: delete some features or use **regularization**"
   ]
  }
 ],
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